Surface
of
spandex
N
q
mg
Gary D. White
National Science Foundation and
American Institute of Physics
gwhite@nsf.gov
4/9/2015
Gulf Coast Gravity Meeting, Oxford
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The Spandex, for
demonstrating celestial phenomena:
• The Solar System
–
–
–
–
–
–
Orbits, precession
Escape velocity
Planetary Rings
Roche Limit
Density differentiation
Early solar system
agglomeration models
• Earth and moon
– Binary Systems
– Tidal effects
•
See ‘Modelling Tidal Effects’, AJP,
April 1993, GDW and students
NOTE: “Gravity wells” rather
than “curved space-time” or
“embedding diagrams”
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Video fun
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From XKCD (A webcomic of romance,
sarcasm, math, and language, http://xkcd.com/681/)
…but are spandex
gravity wells really like
3-D space?
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Wrong things that I thought I knew
about the shape of the Spandex
– “It is like a soap
bubble between
rings.”
Pull middle ring down---this has
long been known to produce a
catenary curve
…or perhaps in
St. Louis, the
curve of the
Arch
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rotate
Better known as the
curve of a hanging
chain
…except
data can’t be
fit to the
appropriate
hyperbolic
cosine… 5
Wrong things that I thought I knew
about the shape of the Spandex
– “Oh, right, it is
like a weighted
drum head.”
M
So, it solves
Laplace’s
equation with
cylindrical
symmetry,
h=A + B*ln(R)
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This would make it
like 2-D gravity, like
orbits around a long
stick of mass M
Gulf Coast Gravity Meeting, Oxford
…except our
original data
couldn’t be
fit to any
logarithmic
form… 6
…until we learned to stretch it
as we attached it
thanks to Don Lemons and TJ Lipscombe, AJP 70, 2002
pho^(2/3)
ln(x)
0
0
0.8
-0.1
-0.2
-0.3
5
4.5
4
log-log plot: unstretched spandex well data
3.5
2.5
3
y = 0.6666x + 0.4895
R² = 0.9971
2
A*ln(x)+B
2.5
1.5
1
2
0.5
1.5
Spandex
1
0
0
0.5
1
1.5
2
2.5
0.5
0
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10
15
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Connection to general relativity
• Wilson (1920!- Phil. Mag. 40, 703) gives the metric for
an infinite wire of mass to be (to leading order in m)
ds   dt  
2
4m
2
8m2
d  2   4mdz 2   2d 2
where m  G / c  [3x10 m / kg ] ; …incredibly
small for any reasonable linear density…in the slow
speed, small mass density limit this means that the the
Newtonian effective potential predicted by Einstein’s
equations of a wire (or long stick or bar galaxy or other
prolate distribution, perhaps) is given by
2
27
Newton  (1  g00 )c2 / 2  (1   4m )c2 / 2  (1  e4mln  )c2 / 2  (2  4m ln   ...)c2 / 2
…In other words, logarithmic, as perhaps expected.
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…so “pre-stretched” Spandex potential well is like the well around a
skinny stick of mass m…but what about rolling marbles on Spandex? Is
that really like planets moving in a logarithmic potential? To relay that
story, let’s recall some of the coolest early science
planets
period, T radius from sun, R T-squared
(in years) (in earth-sun distances)
Mercury 0.241
0.387
0.0580
Venus
0.616
0.723
0.379
Earth
1
1
1
Mars
1.88
1.52
3.54
Jupiter
11.9
5.20
141.6
Saturn
29.5
9.54
870.3
R-squared T-cubed R-cubed
0.150 0.0140
0.523 0.2338
1
1.0000
2.321 6.65
27.1 1685.16
91.0 25672.38
0.058
0.378
1.000
3.54
140.8
867.9
So, in natural units, T2 = R3 for planets.
(In unnatural units, T2 is proportional to R3)
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9
We determined Kepler’s law analog
for unstretched Spandex for circular
orbits by doing some experiments…
• For fixed M, unstretched
Spandex has
•
•
Curiously close, but no cigar;
What is pre-stretched spandex
Kepler’s law analog for circular
orbits?
Let’s come back to that…for now
notice how noisy the data is…
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-6
-2
-4
0
1.5
1
ln(T)
ln(T)=(1/3)ln(R2) +b
– So, Spandex is T3/R2 = k…
– Kepler Law for real
planets about sun is T2/R3
= c.
Kepler's Law analog
0.5
line has slope 1/3
y-intercept ~ 1.35
0
-0.5
-1
ln(R^2/sqrt(M))
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About rolling on the Spandex…
let’s first consider the lower dimensional case---modelling one
dimensional oscillations with motion in a vertical plane
• One-D motion
1
E1D  mVx 2  U  x 
2
Diff. wrt time to get
Rolling in a vertical plane
in a valley given by h(x):
Eroll 
1
1
1
mVx 2  mVy 2  I  2  mgh  x 
2
2
2
mgh(x)
but
Vy  Vx tan(q )  Vx h( x)
0  {mx  U ( x)}Vx
and no-slip rolling means
Assume x  x0   , then
V 2  Vx 2  Vy 2  (a)2
0  m  U ( x0 )  U ( x0 )  ...
so E  1 m 1  I  1  h ( x)V  mgh  x 
q
x
So for small  we get SHM with
0 
U ( x0 )
k

m
m
2
roll
roll 
2

kroll

mroll
2
x
ma 2 
mgh( x0 )
 m  I / a 1  h ( x ) 
2
2
0
Likewise for 2D(that is, when we want to model near circular
2D motion in a plane we can use near-circular motion on a Spandex)…WHY?
First, let’s look at the planar case
E2 D 
1
mV 2  U     L2 / (2m 2 )
2
Diff. wrt time, assume
  R 
0  m  U ( R0 )  U ( R0 ) 
 L2 / (mR03 )  3L2 / (mR0 4 )  ...
U ( R0 )  3L2 / (mR04 )
k
2 Doscillations 

m
m
Again, SHM, constant terms give orbital frequency,
2
L2  mR03U (R0 )  R0 orbital
 U (R0 ) / m  T 2  R0 / U (R0 )  Kepler, etc.
coefficient of  gives frequency of small oscillations about orbit,
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Rolling adds more complications, but when
rolling in a horizontal circle we have something
similar, but with a few new terms due to the
rolling constraint


1
(m  I / a 2 )V 2 (1  h2 )  mgh(  )  1  I (1  h2 ) / (ma 2 ) ( L2 / (2m 2 ) 
2
 Lh / m

2
2

( I / a )(1  h )(a z ) 
 a z 
  1  h2



Erolling 
leading to,


I
cos( )
2
R0 orbital
 gh0 / 1 

2
ma
cos(
q
)
cos(


q
)


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Gulf Coast Gravity Meeting, Oxford
instead of Kepler’s Law
2
R0 orbital
 U (R0 ) / m
13
R0
2
orbital


I
cos( )
 gh0 / 1 

2
ma
cos(
q
)
cos(


q
)


• For   q
2
 gh0
have R0 orbital
• So if h(R) is power law,
h( R)  A R
or
T
I

 1 
R  ma 2
2
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  2 
2
 
 gh ( R )
 1
yielding Kepler’s law analog
2 
R
I  2

 1 
=constant
2
2 
T
 ma  gA
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Effect of rolling on orbits
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Returning to the question of
what is the pre-stretched Spandex vesion of Kepler’s Law
R0
• For
have
2
orbital


I
cos( )
 gh0 / 1 

2
ma
cos(
q
)
cos(


q
)


 q
2
R0 orbital
 gh0
or
T
I

 1 
R  ma 2
2
  2 
 
 gh ( R )
2
• So if h(R) is logarithmic,
h( R)  A / R
yielding
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2 R
 V  constant
T
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What if not going in a circle?
• For cones, the oscillations about near circular
motion satisfy

2
rolling oscillations
 I (1  h2 ) 
I
 3 1 

2
 (az / R0 )h / (1  h2 )3/2

2
2
ma
ma


2
• Can also derive neat analytical expression for
“scattering angle” for cones…

• Spandex is more
 
1  sin 2 (q ) / B
complicated…
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Another video, ball in cone
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Moving in a cone---exp. vs theory
for near circular orbits
Torbit exp  0.97s
Tdevexp  0.72s
exp  6.48  .2rad / s
exp  8.73  .4rad / s
theory  8.96  .4rad / s
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Two comments
1) Should imperfect
models, like Spandex
and cones be used to
convey ideas about
gravity, general
relativity?
• Yup,
(Imperfect models are
better than “perfect”
ones (consider “fullscale” maps!))
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2) Recall that to get
logarithmic potential that
is like real gravity for
wire-shaped mass
distributions, we have to
stretch the Spandex taut
and then add a heavy
mass. Why is that?
…Why do you have to
stretch the Spandex for it
to model the real gravity?
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Was real gravity prestretched?
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Thanks to
• My students, especially Michael Walker, Tony
Mondragon, Dorothy Coates, Darren
Slaughter, Brad Boyd, Kristen Russell, Matt
Creighton, Michael Williams, Chris Gresham,
Randall Gauthier.
• Society of Physics Students (SPS) interns
Melissa Hoffmann and Meredith Woy
• Aaron Schuetz, Susan White,
• SPS staff, AIP, APS, AAPT, NSF, NASA, and
• You!
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